Problem 12 - Fundamentals of Mathematics

Question

Find each value. \(|-5|+|-10|\)

Step-by-Step Solution

Answer

The value is 15.

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, the absolute value of any number is positive.

Step 2: Calculate the Absolute Value of -5

Compute \(|-5|\). The distance from 0 is 5, hence \(|-5| = 5.\)

Step 3: Calculate the Absolute Value of -10

Compute \(|-10|\). The distance from 0 is 10, hence \(|-10| = 10.\)

Step 4: Add the Absolute Values Together

Add the absolute values from the previous steps: \(5 + 10 = 15.\)

Key Concepts

Understanding the Number Line

The number line is a fundamental concept in mathematics that helps us understand the relative positions of numbers. Imagine a straight horizontal line where numbers are placed at equal intervals. Zero is situated at the center, with positive numbers extending to the right and negative numbers extending to the left. This visual representation makes it easier to comprehend concepts like absolute value.

• Numbers to the right of zero are positive.
• Numbers to the left of zero are negative.
• The distance a number is from zero determines its absolute value.

Understanding the number line allows you to easily see how far a number is from zero. Absolute value then becomes simply a measure of this distance without worrying about direction. This understanding aids learning in mathematical operations and solving equations easily.

Exploring Positive Numbers

Positive numbers are the integers located to the right of zero on the number line. They are greater than zero and are used in various mathematical operations such as addition, multiplication, and more. In real life, they often represent quantities, such as counting numbers, or measurements, like lengths and distances.

• Examples include 1, 2, 3, and 4.
• Positive numbers are endless in quantity and increase without bound as they extend to the right on the number line.
• They are crucial for expressing gains or profits in finance, heights above sea level in geography, and other real-world situations.

Understanding positive numbers helps in performing everyday calculations and grasping concepts like absolute value, which always results in a positive number, reflecting the value's distance from zero.

Absolute Value in Mathematics Education

In mathematics education, absolute value reinforces the understanding of distance and magnitude without worrying about direction. It is a non-negative value that signifies how far a number is from zero on the number line, regardless of whether the original number is positive or negative.

• Absolute value is symbolized by vertical bars, for example, \(|x|\).
• It helps simplify expressions and computations, especially when dealing with equations and inequalities.
• Learning absolute value is an essential step in developing a student's numerical fluency and problem-solving skills.

In mathematics education, exploring concepts like absolute value supports broader mathematical understanding, aiding students in engaging with more complex mathematical theories and real-world applications effectively.